Embedded and immersed boundary methods have been for decades for approximating boundary value problems (BVP) in domains with under resolved boundaries. Different strategies exist. The classical Brinkman penalisation is quite robust but has strong accuracy limitations. The immersed ghost fluid method allows to meet the physical boundary condition within a given accuracy on the immersed surface. It is however known to generate instabilities when the physical boundary is too close to the mesh nodes. Other methods, as the shifted boundary method (SBM), rely on extrapolation strategies from the physical to a mesh boundary do not have such drawbacks, and overcome the accuracy limitation. However, in some cases surprising constraints need to be included to avoid spurious modes. The aim of this work is to provide a different point of view of these high order methods by means of a notion of sub-grid corrections inpsired by asymptotic expansions of the solution. By looking at the expansion of the operator within a thin layer enveloping the mesh boundaries closest to the embedded surface we deduce the appropriate form of corrections to the solution compatible with the accuracy sough. This also allows to provide some insights in the stability of the resulting method, to justify some of the constructions proposed in literature, and to propose some new approaches.